please see attached document. need step by step typed solutions! thank you!

 

 

YOU MUST SHOW APPROPRIATE WORK TO JUSTIFY YOUR ANSWERS/SOLUTIONS! YOU MUST SHOW APPROPRIATE WORK TO JUSTIFY YOUR ANSWERS/SOLUTIONS!
I. [12 + 6 + 6 = 24 points]
i. Given: tan( xy 2 ) e x y , find dy/dx by implicit differentiation.
2 ii. Use formulas for derivatives of inverse trig functions to compute:. d
cot 1 ( x) tan 1 (1 / x)
dx YOU MUST SHOW APPROPRIATE WORK TO JUSTIFY YOUR ANSWERS/SOLUTIONS! iii. Compute y using the formulas for the inverse trig functions if y sin 1 cos 1 (e x ) .
2 [12 + 5 = 17 points] x 2 3 / 4 . ecos x . tan 6 x
dy
y II. i. Given
, use logarithmic differentiation to evaluate .
3 4
dx
1 x YOU MUST SHOW APPROPRIATE WORK TO JUSTIFY YOUR ANSWERS/SOLUTIONS! e7 x e 7 x
x e7 x e 7 x ii. Compute the limit: lim III. [10 + 6 + 6 = 22 points]
i. Use an appropriate local linear approximation to estimate 5.97 5 / 4 correct up to three decimal places. YOU MUST SHOW APPROPRIATE WORK TO JUSTIFY YOUR ANSWERS/SOLUTIONS! ii. Compute y if y ln ln ln ln sin(2 x) iii. Find the domain of the function: f ( x) x e2x 2 e1 x e 2 x 2 . Write your answer in interval notation. Use the exact values-no decimal approximations! IV. [16 points]
A spherical balloon is to be deflated so that its radius decreases at a constant rate of 15 cm/min. At what
rate must air be removed (hint: volume decreases) when the radius is 10 cm?
i. Clearly declare your variables.
ii.Identify the given and the asked rates.
iii. Set up an equation that relates the variables whose rates of change have been identified.
iv. Finish solving the problem. YOU MUST SHOW APPROPRIATE WORK TO JUSTIFY YOUR ANSWERS/SOLUTIONS! V. [14 + 7 = 21points]
i. A radioactive substance has a starting mass of 100 mg, which is reduced to 90 mg in 7 days.
a. Find a formula for the mass m(t) of this radioactive substance after t days.
b. Compute the half-life (i.e., time it takes the 50% of the substance to decay) of this radioactive
substance. YOU MUST SHOW APPROPRIATE WORK TO JUSTIFY YOUR ANSWERS/SOLUTIONS! ii. Compute the limit: lim x ln 2 x
ln 3 x A 13 ft ladder is leaning against a wall. If the bottom of the ladder slides away from the wall at a constant rate
of 2 ft/sec, how fast will the top of the ladder be sliding down the wall when the bottom of the ladder is 8 ft
away from the wall?

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